## Interpolated and warped 2-D digital waveguide mesh algorithms (2000)

Venue: | in COST G-6 Conference on Digital Audio Effects |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Välimäki00interpolatedand,

author = {Vesa Välimäki and Lauri Savioja},

title = {Interpolated and warped 2-D digital waveguide mesh algorithms},

booktitle = {in COST G-6 Conference on Digital Audio Effects},

year = {2000},

pages = {7--9}

}

### OpenURL

### Abstract

Interpolated and warped digital waveguide mesh algorithms have been developed to overcome the problem caused by direction and frequency-dependence of wave travel speed in digital waveguide mesh simulations. This paper reviews the interpolation methods applicable in the two-dimensional case. The bilinear interpolation technique and two other approaches are briefly recapitulated. The use of 2-D quadratic interpolation results in a mesh structure that is efficiently implemented using only additions and binary shifts. The optimized interpolated 2-D mesh is obtained by choosing the free parameters of the interpolation method so that the difference in wave travel speed in the axial and diagonal directions is minimized. The output signal of any digital waveguide mesh can be processed with a warped FIR filter to reduce the dispersion error at all frequencies. The frequency-warping factor must be optimized individually for each mesh structure. We introduce an extended method called multiwarping, which can be used to further decrease the dispersion error. The frequency range of digital waveguide mesh simulations is also discussed. 1.

### Citations

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(Show Context)
Citation Context ...er decrease the dispersion error. The frequency range of digital waveguide mesh simulations is also discussed. 1. INTRODUCTION The digital waveguide mesh was introduced in 1993 by Van Duyne and Smith =-=[1, 2]-=-. They showed that the method is suitable for sound synthesis of percussion instruments although it suffers from direction-dependent dispersion. In 1994, Savioja et al. extended the use of the digital... |

49 |
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Citation Context ... frequency of the rectangular waveguide mesh, which is 0.25. ξ 2 c 0.2 0.1 −0.1 −0.2 existing neighboring nodes to obtain a realizable structure. This is a 2-D application of fractional delay filters =-=[15]-=-. The method of superimposing signal samples onto a non-integer point of a digital delay line was introduced in [16], where this operation was named as ‘deinterpolation’. See also [17, pp. 128–135], w... |

49 | Discrete-time modeling of acoustic tubes using fractional delay filters - Välimäki - 1995 |

47 |
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- Härmä, Karjalainen, et al.
- 2000
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Citation Context ...rocessed with it. 3.1. How to Use Frequency Warping Frequency warping can be implemented with a warped FIR filter, which consists of a chain of first-order allpass filters A(z), as shown in Fig. 4(a) =-=[7, 6, 9]-=-. The tap coefficients are set equal to the signal samples x m(n) to be warped, i.e., the output signal of the digital waveguide mesh in this case. When a unit impulse is fed into the filter, the warp... |

45 |
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Citation Context ...rocessed with it. 3.1. How to Use Frequency Warping Frequency warping can be implemented with a warped FIR filter, which consists of a chain of first-order allpass filters A(z), as shown in Fig. 4(a) =-=[7, 6, 9]-=-. The tap coefficients are set equal to the signal samples x m(n) to be warped, i.e., the output signal of the digital waveguide mesh in this case. When a unit impulse is fed into the filter, the warp... |

42 |
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Citation Context ...rom direction-dependent dispersion. In 1994, Savioja et al. extended the use of the digital waveguide mesh to three dimensions, and presented simulation results of wave propagation in acoustic spaces =-=[3]-=-. The method has turned out to be useful in acoustic design of listening rooms, for example. Recently, the digital waveguide mesh has been used in the modeling of drums [4] and the violin body [5]. Th... |

38 |
The 2-D Digital Waveguide Mesh
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(Show Context)
Citation Context ...er decrease the dispersion error. The frequency range of digital waveguide mesh simulations is also discussed. 1. INTRODUCTION The digital waveguide mesh was introduced in 1993 by Van Duyne and Smith =-=[1, 2]-=-. They showed that the method is suitable for sound synthesis of percussion instruments although it suffers from direction-dependent dispersion. In 1994, Savioja et al. extended the use of the digital... |

36 |
Reducing the dispersion error in the digital waveguide mesh using interpolation and frequency warping techniques
- Savioja, Välimäki
- 2000
(Show Context)
Citation Context ... mesh has been used in the modeling of drums [4] and the violin body [5]. The interpolated mesh was developed to overcome the direction-dependent wave propagation characteristics and dispersion error =-=[6]-=-. However, it was only possible to reduce the directiondependence while the dispersion was not much affected—it was merely rendered almost independent of direction. Luckily, the remaining dispersion e... |

28 | The 3D Tetrahedral Digital Waveguide Mesh with Musical Applications - VanDuyne, Smith - 1996 |

22 | Physical modeling of membranes for percussion instruments - Fontana, Rocchesso - 1998 |

20 | Signal-theoretic characterization of waveguide mesh geometries for models of two dimensional wave propagation in elastic media
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(Show Context)
Citation Context ...y range of digital waveguide mesh simulations. Fontana and Rocchesso have discussed earlier that the triangular mesh is superior over rectangular or hexagonal tilings in terms of valid frequency band =-=[20]-=-. Figure 5 shows the mapping of the original normalized frequencies to the frequencies occurring on the mesh in the case of the original (Fig. 5(a)) and optimally interpolated meshes, when MESH FREQUE... |

18 |
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Citation Context ...s transfer functions A(z) instead of the first-order one. However, replacing unit delays with second or higher order allpass filters leads to both folding and warping in the delay-line implementation =-=[18]-=-, which is undesirable. Another attractive solution would be to apply the first-order warping many times successively, but this is a waste of effort since there will not be more degrees of freedom in ... |

15 | A new formulation of the 2-D waveguide mesh for percussion instruments - Fontana, Rocchesso - 1995 |

11 |
Modelling a Drum by Interfacing 2-D and 3-D Waveguide Meshes
- Aird, Fitch
(Show Context)
Citation Context ...pagation in acoustic spaces [3]. The method has turned out to be useful in acoustic design of listening rooms, for example. Recently, the digital waveguide mesh has been used in the modeling of drums =-=[4]-=- and the violin body [5]. The interpolated mesh was developed to overcome the direction-dependent wave propagation characteristics and dispersion error [6]. However, it was only possible to reduce the... |

10 |
Reduction of the dispersion error in the triangular digital waveguide mesh Vol. 32. Pt. 5. 2010 of the Institute of Acoustics using frequency warping
- Savioja, Valimaki
- 1999
(Show Context)
Citation Context ...n that the frequencywarping can be incorporated in the mesh structure [10]. The best results so far have been obtained by using the triangular waveguide mesh [11–13] together with a warped FIR filter =-=[14, 6]-=-. Also, the interpolated rectangular mesh can be improved considerably using frequency warping [6]. In Section 2 of this paper, we review the interpolation methods for the rectangular 2-D digital wave... |

9 | Warped filters and their audio applications - Karjalainen, Hrm, et al. - 1997 |

8 |
Fractional delay digital filters
- VŠlimŠki, Karjalainen, et al.
- 1993
(Show Context)
Citation Context ...btain a realizable structure. This is a 2-D application of fractional delay filters [15]. The method of superimposing signal samples onto a non-integer point of a digital delay line was introduced in =-=[16]-=-, where this operation was named as ‘deinterpolation’. See also [17, pp. 128–135], where deinterpolation is discussed in detail. In the initial version of the interpolated mesh algorithm [6], four hyp... |

5 | A waveguide mesh model for high-frequency violin body resonances
- Huang, Serafin, et al.
- 2000
(Show Context)
Citation Context ...ces [3]. The method has turned out to be useful in acoustic design of listening rooms, for example. Recently, the digital waveguide mesh has been used in the modeling of drums [4] and the violin body =-=[5]-=-. The interpolated mesh was developed to overcome the direction-dependent wave propagation characteristics and dispersion error [6]. However, it was only possible to reduce the directiondependence whi... |

5 |
Improving the three-dimensional digital waveguide mesh by interpolation
- Savioja
- 1998
(Show Context)
Citation Context ...n the two-dimensional digital waveguide mesh methods, the interpolation and frequency-warping methods are applicable to three-dimensional simulations as well. Promising results have been published in =-=[21]-=-, where the interpolation technique was adapted to the 3-D rectangular digital waveguide mesh. 7. ACKNOWLEDGMENTS The work of V. Välimäki has been supported by a postdoctoral research grant from the A... |

4 | Online correction of dispersion error in 2D waveguide meshes
- Fontana, Rocchesso
(Show Context)
Citation Context ... It is implemented by postprocessing the output signal of the mesh using a warped FIR filter. Fontana and Rocchesso have also shown that the frequencywarping can be incorporated in the mesh structure =-=[10]-=-. The best results so far have been obtained by using the triangular waveguide mesh [11–13] together with a warped FIR filter [14, 6]. Also, the interpolated rectangular mesh can be improved considera... |

4 |
Multiwarping for enhancing the frequency accuracy of digital waveguide mesh simulations
- Savioja, Välimäki
- 2001
(Show Context)
Citation Context ... the frequency warping obtained with two successive warping operations with λ 1 and λ 2 is equivalent to warping with (λ 1 + λ 2) / (1 + λ 1λ 2). We propose a new approach that we call ‘multiwarping’ =-=[19]-=-, where an arbitrary number of successive first-order frequency warping and resampling operations are performed alternately as illustrated in Fig. 4(b). The use of sampling rate conversions facilitate... |